Observational properties of rigidly rotating dust configurations

We study the observational properties of a class of exact solutions of Einstein’s field equations describing stationary, axially symmetric, rigidly rotating dust (i.e. non-interacting particles). We ask the question whether such solutions can describe astrophysical rotating dark matter clouds near the center of galaxies and we probe the possibility that they may constitute an alternative to supermassive black holes at the center of galaxies. We show that light emission from accretion disks made of ordinary baryonic matter in this space-time has several differences with respect to the emission of light from similar accretion disks around black holes. The shape of the iron K\(\alpha \) line in the reflection spectrum of accretion disks can potentially distinguish this class of solutions from the Kerr metric, but this may not be possible with current X-ray missions.

Observational properties of rigidly rotating dust configurations

Eur. Phys. J. C
Observational properties of rigidly rotating dust configurations
Batyr Ilyas 2
Jinye Yang 1
Daniele Malafarina 2
Cosimo Bambi 0 1
0 Theoretical Astrophysics, Eberhard-Karls Universität Tübingen , Auf der Morgenstelle 10, 72076 Tübingen , Germany
1 Center for Field Theory and Particle Physics and Department of Physics, Fudan University , 220 Handan Road, 200433 Shanghai , China
2 Department of Physics, Nazarbayev University , 53 Kabanbay Batyr avenue, 010000 Astana , Kazakhstan
We study the observational properties of a class of exact solutions of Einstein's field equations describing stationary, axially symmetric, rigidly rotating dust (i.e. noninteracting particles). We ask the question whether such solutions can describe astrophysical rotating dark matter clouds near the center of galaxies and we probe the possibility that they may constitute an alternative to supermassive black holes at the center of galaxies. We show that light emission from accretion disks made of ordinary baryonic matter in this space-time has several differences with respect to the emission of light from similar accretion disks around black holes. The shape of the iron Kα line in the reflection spectrum of accretion disks can potentially distinguish this class of solutions from the Kerr metric, but this may not be possible with current X-ray missions.
1 Introduction
We know that supermassive compact objects exist at the
center of galaxies [
1
]. It is usually believed that they must be
black holes even though at present we do not have any
conclusive evidence about their nature. For this reason it is worth
investigating the possibility that these sources may not be
black holes and if there are possible observational tests that
can distinguish between black holes and other, more exotic,
sources of the gravitational field [
2–5
]. The general theory
of relativity allows for several exact solutions that do not
describe black holes and that could be used to model
gravitating compact objects. Indeed in the last few years there has
been great interest in the study of the motion of test
particles on accretion disks around exotic compact objects and
naked singularities and in the possible ways to distinguish
such space-times from black holes (see for example [
6–8
]).
The study of the disk’s reflection spectrum is a
promising tool to observationally test the space-time metric around
these supermassive objects [
9–12
]. The most prominent
feature is the iron Kα line, which is very narrow in the rest
frame of the gas and is instead broad and skewed in the
observed spectrum of a black hole candidate due to
relativistic effects (gravitational redshift, Doppler boosting, light
bending) occurring in the strong gravity region. In the
presence of high quality data and with the correct astrophysical
model, this approach can provide stringent constraints on the
nature of astrophysical black hole candidates.
In Ref. [
13
], two of us studied the shape of the iron Kα
line coming from accretion disks inside low density matter
clouds with a central singularity and compared with the same
line produced by accretion disks around black holes. In the
present article, we focus on rotating non-vacuum solutions as
possible candidates for supermassive compact objects. This
is an ideal natural continuation of the work presented in
Ref. [
13
], as here we consider an axially symmetric matter
source with non-vanishing angular momentum.
The model is made of three elements: (i) As the matter
source that determines the geometry we consider a cloud of
rigidly rotating dust (i.e. non-interacting particles). (ii) Due to
the divergence of the energy density at the center we consider
a cutoff of the metric at a minimum radius r0 and assume the
existence of an exotic compact object at the center. (iii) In
order to test the observable features of the space-time we
consider a luminous accretion disk made of ordinary matter
in the equatorial plane and we compare its features with the
features of similar disks around black holes.
Stationary, axially symmetric space-times are of great
importance in astrophysics as they may be connected to the
origin of extragalactic jets. For example, in Ref. [
14
], it was
shown that rigidly rotating dust configurations present a
density gradient parallel to the axis, which is missing in the
corresponding Newtonian case. The argument, put forward by
Opher, Santos and Wang in Ref. [
15
] is that the space-time
geometry of a rotating dust cylinder, such as that presented in
Ref. [
16
], can contribute to the collimating effects observed in
extragalactic jets. Unfortunately, when considering rotating
fluids in general relativity (GR), there are several pathologies
that may arise, such as singularities, non asymptotic flatness
and negative densities, and these may hinder our
understanding of the physical significance of the source.
In the present article, we study accretion disks in the
interior of a fluid body made of weakly interacting particles. One
can think of such a pressureless fluid which does not interact
with ordinary particles as describing the central region of a
dark matter cloud. To this aim, we will focus on an exact
solution describing a rigidly rotating dust cloud that was first
discussed by Bonnor in Ref. [
14
]. The Bonnor solution is a
member of the family of stationary, axially symmetric
spacetimes that was derived by Winicour in Refs. [
17,18
]. This
solution is particularly appealing because it is
asymptotically flat and its energy density is everywhere positive, so it
is physically viable, except for its center, where a negative
mass singularity is present. Also a dust source that gravitates
but does not interact with the particles in the accretion disk
can be interpreted as a rotating cloud of dark matter. Our aim
here is to investigate whether such fluid can be used to model
an astrophysical supermassive compact object and if such a
hypothetical source can be distinguished from a black hole
via the emission spectrum of its accretion disk.
The article is organized as follows. In Sect. 2, we briefly
discuss stationary axially symmetric solutions of Einstein’s
field equations and in particular rigidly rotating dust. In
Sect. 3, we describe the properties of accretion disks in such a
solution and compare the luminosity spectrum emitted by the
disk with that coming from a black hole. In Sect. 4, we
simulate the emission of the iron Kα line from such accretion disks
and discuss whether such lines can be detected in
observations of supermassive compact objects. Section 5 is devoted
to discussing the results and future directions. Throughout
the article we make use of natural units setting G = c = 1.
2 Rotating dust
The metric for a stationary, axially symmetric rigidly rotating
dust source was found by Van Stockum (see [
19–21
]) and we
can write it in cylindrical coordinates {t, ρ , φ, z} as
ds2 = −dt 2 + 2ηdt dφ + eμ(dρ2 + dz2) + (ρ2 − η2)dφ2,
where η(ρ , z) and μ(ρ , z) are the metric functions to be
determined via Einstein’s equations. If we define a function
ξ(ρ, z) from
where ∇2 indicates the Laplace operator in the flat
threedimensional space. Once a solution of Eq. (3) is given,
Eqs. (4) and (5) are immediately integrated by quadrature.
Therefore the whole problem of finding a solution of
Einstein’s equations describing stationary, axially symmetric,
dust reduces to the problem of solving Laplace’s equation
in ordinary flat three-dimensional space. Note that the same
is true for static, vacuum, axially symmetric solutions, and
therefore there exists a one to one correspondence between
static, vacuum, axially symmetric solutions and stationary,
axially symmetric rigidly rotating dust. Einstein’s equations
for dust take the form
1
Ri j − 2 gi j R = 8π ui u j ,
8π
e−μ
= ρ2 (η,2ρ + η,2z ).
where ui is the four-velocity (given by ui = δ0i) and (ρ , z)
is the energy density of the dust cloud and it is given by
η(ρ , z) = ρξ,ρ ,
then the problem reduces to solving
∇2ξ = 0,
2 2
η,z − η,ρ ,
μ,ρ = 2ρ
η,z η,ρ ,
μ,z = − ρ
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
Since we are considering stationary, axially symmetric
configurations we will look for a solution of Eq. (3) that
does not depend on t and on the angular variable φ. Also,
the fact that the metric is stationary and axially symmetric
implies that we have the usual commuting Killing vectors
T μ = δtμ and μ = δφμ. The four-velocity is then given
by uμ = T μ + 0 μ, where 0 is the (constant) angular
velocity.
2.1 Bonnor’s solution
The simplest solution of Eq. (3) can be obtained taking
2h
ξ = r ,
where we have set r = ρ2 + z2 and h is an arbitrary
constant. This is the space-time first investigated by Bonnor in
[
14
]. For the metric functions we get
(11)
and the corresponding energy density is given by
8π
=
For this model it is relatively easy to evaluate μ and η and at
spatial infinity the metric tends to become Minkowski in all
directions. The energy density is always positive and it drops
rapidly as r increases but on the other hand it presents a
singularity at r = 0. The total mass of the system is zero due to
a distributional infinite negative mass located at the
singularity, which balances the positive mass and makes the metric
somehow unphysical. Nevertheless, thinking about an
astrophysical source, the presence of a singularity at the center
indicates a regime where classical general relativity does not
hold anymore. In fact, diverging quantities typically suggest a
breakdown of the model and in particular singularities in GR
are generally associated with a failure of the classical theory
to describe strong gravitational fields over short distances.
In order to have a complete description of the behaviour of
gravity in these extreme conditions one would need to use
a theory of quantum gravity. However, despite the lack of
a viable quantum gravity theory at the moment, there exist
several indications of the possible corrections that such a
theory would induce on classical models (see Ref. [
22
] and
the references therein). Therefore it is reasonable to argue
that the singularity at the center of the Bonnor space-time
would be resolved by a theory of quantum gravity and we
can assume that the present solution will be valid from a
certain radius r0 outwards, while the behaviour for r < r0
cannot be properly captured by the classical solution. We can
further speculate that the resolution of the singularity may
imply the existence of an exotic compact object at the
center. Exotic compact objects have been proposed for decades
also within classical GR. The size and properties of such
objects vary from model to model. For example, objects like
quark stars [
23
] and boson stars [
24
] are slightly bigger than
the Schwarzschild radius for a black hole of the same mass,
while gravastars [
25
] for example have a radius of the order
of the Schwarzschild radius. Other, more compact, objects
have been suggested as well (see for example Ref. [
26
]) and
they may be intrinsically quantum in nature.
In this case, we assume that we can neglect the finite
amount of matter enclosed between the center and r0 and
evaluate the total positive mass of the system as the amount
of matter contained between r0 and infinity. We find
3π h2
M0 = 16r03 .
Note that as r0 goes to zero the mass diverges, while the
total mass for the system is zero. This suggests that the
central singularity is acting as a distribution of infinite negative
mass that balances the positive mass of the space-time. We
shall mention here that another way to deal with the presence
(12)
(13)
(14)
(15)
(16)
of the singularity is by constructing a solution describing a
rigidly rotating dust configuration accompanied by a rigidly
rotating thin disk located on the equatorial plane (see for
example Ref. [
27
] for a relativistic rigidly rotating disk of
dust in vacuum). The metric then results from the matching
of three parts, two rigidly rotating dust space-times one above
and one below the equatorial plane, and the disk. The matter
distribution on the disk is then given by jump conditions for
the second fundamental form across the disk. Such a model
effectively removes the plane where the singularity is located
and replaces it with another matter source. The properties of
this kind of models are beyond the scope of the present article
and will be investigated elsewhere.
2.2 Rotating dust as dark matter
As we mentioned before our aim is to investigate the
possibility that relativistic rotating dust could be used to describe dark
matter clouds at galactic centers, possibly without invoking
the presence of supermassive black holes. Dark matter clouds
in galaxies can be described by Newtonian density profiles
that give a phenomenological account of the missing matter
distribution that is necessary to explain the velocity
dispersion of stars in the outer regions of galaxies.
One of the most widely used density profiles is the
Navarro–Frenk–White (NFW) model for which the density
takes the form
ε =
r
rs
ε0
1 + rs
r 2
,
where ε0 is a constant describing the characteristic density
of the model and rs is a parameter related to the scale [
28
].
The NFW profile is a variation of the Jaffe profile [
29
], given
by
which was originally derived by observing the brightness of
spiral galaxies. Other density profiles that can be considered
are the pseudo-isothermal sphere [
30
], given by
ε =
ε =
ε =
r 2
rs
ε0
1 + rs
r 2
,
ε0
1 + rs
r 2
,
ε0
r
1 + rs
1 + rs
r 2
or the universal rotation curve profile [
31
], given by
As said, these profiles are constructed in order to explain
the observed behaviour of galaxies and galaxy clusters at
large scales and they need not provide a good description
for dark matter near the center. In fact some profiles, like
the NFW for example, present cusps or divergencies at the
center.
The density profile obtained from the Bonnor solution is
also not well behaved at the center (as it goes like 1/r 6 on the
equatorial plane, near the center), presenting an even steeper
divergence with respect to the other profiles. On the other
hand the Bonnor profile differentiates itself from the
standard profiles mentioned above in the fact that it is axially
symmetric and it includes the effects of rotation (notice that
both features may be testable in principle). Of course this
need not be the best description for dark matter near galactic
centers as there may be more realistic models, which
possibly take into consideration a transition from a Newtonian
profile at large distances to a relativistic profile near r = 0.
However, the main point that we want to stress here is that
relativistic effects due to the high density reached towards
the inner region of the galaxy may play an important role
and affect the dynamics of accreting gas. In this case a
simple analytical relativistic model like the Bonnor space-time
does provide indications on the general behaviour of
ordinary matter moving inside a dense but pressureless fluid and
allows for comparison with the motion of particles in a
vacuum black hole space-time.
2.3 Other solutions
As we have said, to every solution of Laplace equation
corresponds a rigidly rotating dust space-time. Therefore other
matter sources can be found by solving Eq. (3) via separation
of variables with ξ not depending on φ. If we consider the
simplest non-trivial solution given by
(17)
ξ (ρ , z) = J0(ρ)e−α|z|,
where α is a positive constant and J0(ρ) is the zero order
Bessel function of the first kind, we can obtain a density
profile that does not diverge anywhere. This solution was studied
in Ref. [
32
] in connection to the possible appearance of jets
powered by a non-vanishing density gradient along the axis.
In this case the expressions for η and μ become rather
complicated and involve Bessel functions. The energy density
for this model can be calculated from Eq. (7) and it is finite
along the central axis while again it decreases rapidly away
from the axis. Note, however, that the above solution is not
asymptotically flat. It is known, for example, that a
spherically symmetric dust cloud, matched to a vacuum
asymptotically flat exterior must necessarily collapse. Similarly in
order to prevent a rigidly rotating dust solution from
collapsing one has to introduce some other effects. In the case of
the Bonnor solution it is the negative mass distribution at the
singularity. For rotating dust solutions with a regular density
profiles such as the one given by Eq. (17), in order to have
a stationary configuration one must allow for some energy
distribution at infinity. In this case the space-time will be
asymptotically anti-DeSitter, to balance the positive energy
of the dust profile that would cause the whole configuration
to collapse.
3 Accretion disks
In this section we study the motion of test particles moving
on circular geodesics in the equatorial plane of the Bonnor
space-time. These particles represent particles in the
accretion disk of the space-time, while the dust particles of the
source represent a dark matter cloud that extends
throughout the galaxy and has higher density towards the center. So
we model a dark matter cloud near the center of a galaxy
by assuming that we can have circular geodesics inside the
source, and that test particles will move on such geodesics
unaffected. Dark matter has detectable gravitational effects
but does not interact, or interacts very weakly, with ordinary
matter. For this reason we consider dust as a viable theoretical
model for dark matter.
If we pass to spherical coordinates {r, θ } via the
transformation z = r cos θ the metric takes the form
ds2 = −dt 2 + 2ηdt dφ + eμ(dr 2 + r 2dθ 2)
+(r 2 sin2 θ − η2)dφ2,
(18)
where now η and μ are functions of r and θ . For an observer
at infinity, the Bonnor space-time is equivalent to that of a
spinning massless particle located at the origin. In order to
describe test particles in the accretion disk, we will restrict
our analysis to the equatorial plane thus taking θ π/2.
Given the existence of the two killing vectors associated
with time translations and spatial rotations we can define two
conserved quantities, namely the energy E and the angular
momentum L for particles in the accretion disk. Then we
can express the effective potential for a test particle in the
equatorial plane of the Bonnor space-time as
VBeoffnn = 1 −
= 1 − E 2
E 2 gφφ + 2E Lgtφ + L 2 gtt
2
gtφ − gtt gφφ
L 2 4E L h
+ r 2 + r 3 + r 4
4E 2h2
Note the absence of the term in 1/r as opposed to the
Schwarzschild or Kerr case. This is due to the fact that the
only source of gravity for the Bonnor space-time is angular
momentum (the total mass being zero). Also note that, in
order to have a bound motion in the equatorial plane, we
must take h < 0. This can be understood again as a result of
the absence of an attractive positive mass, thus implying that
all the gravitational effects are due to the angular momentum
of the source. For comparison the effective potentials for
Schwarzschild and Kerr are given by
VSecffhw = 1 − E 2
VKefefrr = 1 − E 2
2M
− r
2M
− r +
,
From the condition for circular motion, given by Veff (r ) =
Veff,r(r ) = 0, we see that
E = −
L =
gtt + gtφ
−gtt − 2gtφ
− gφφ
gtφ + gφφ
−gtt − 2gtφ
− gφφ
,
where is the angular velocity of the particles in the
accretion disk and it is given by
dφ
= dt =
2
gtφ,r − gtt,r gφφ,r − gtφ,r
gφφ,r
Circular orbits can exist only if gtt + 2gtφ
thus only for − < < +, where
2|h|r
= r 4 + 4h2
+ gφφ
± = ω ±
ω2
gtt ,
− gφφ
and ω = −gtφ /gφφ is the frame dragging frequency of the
space-time. The limiting case for which = ± defines
the photon capture spheres of the space-time. It is easy to
see from Fig. 1 that as particles get closer to the center
the behaviour of the effective potential differs drastically
between the Bonnor model and the Schwarzschild or Kerr
black holes. This suggests that the emission spectrum of such
accretion disks will also be considerably different.
The next step is to compare the luminosity flux of the
accretion disk in the Bonnor space-time with that of a black
hole. The luminosity flux per unit accretion mass is given by
f (r )
m
˙
ω,r
= − √−g(E − ω L )2
r
rISCO
(E − ω L )L ,r dr,
(26)
where m˙ is the mass accretion rate onto the central object
and rISCO is the radius of the innermost stable circular orbit
(ISCO). Usually the ISCO, which marks the boundary of the
accretion disk, is defined by the condition that Veff,rr = 0 and
for Schwarzschild, for example, it is located at three times
the horizon radius. As for the Bonnor space-time it is easy
to verify that there is no ISCO, meaning that particles are
allowed to move in circular orbits all the way to the central
singularity. This is similar to the behaviour found in other
space-times describing non-vacuum interiors (see for
example [
13
]) and therefore it is natural to take another parameter
as the boundary of circular orbits. For the Bonnor space-time
we take r0 = rISCO and we evaluate the luminosity flux per
unit accretion as
V eff
V eff
Fig. 1 Top panel Comparison of the effective potential between two
models of the Bonnor solution (dashed line with h = −0.000214 and
dot-dashed line with h = −10) with the Schwarzschild case
(continuous line) with M = 1. Chosen values for energy and angular momentum
are E = 1.01 and L = √24. Bottom panel Comparison of the effective
potential between the Bonnor model with h = −0.000214 (continuous
line) and Kerr solutions with M = 1 and angular momentum parameter
a = 0.01 (dot-dashed line) and a = 0.9 (dashed line). Chosen values
for energy and angular momentum are E = 1.01, L = √24
f (r )
m
˙
2√2|h|(4h2 − 3r 2)
= eh2/2r 4 (4h2 + r 4)√r 4 − 4h2
× tan−1 √r − 21 ln √√rr −+ 11
+ C (r0) ,
(27)
where C (r0) in an integration constant that depends on the
choice of r0. The plot of the luminosity is in Fig. 2. As
expected, given the presence of the singularity, the total
luminosity flux for the Bonnor space-time is diverging. Making
the assumption that the singularity is removed we can still
notice that the total luminosity given away by such an object
would be much greater than the luminosity emitted by the
accretion disk surrounding a Schwarzschild black hole. This
is in agreement with what was found in [
6
] and suggests the
possibility that very luminous sources may be less massive
exotic compact objects rather than very massive black holes.
4 Iron Kα line
Within the disk-corona model, a black hole is surrounded by
a geometrically thin and optically thick accretion disk that
0.25
0.2
x
luF0.15
n
o
t
hoP 0.1
Fig. 2 Top panel Comparison between the luminosity flux from
accretion disks around a Schwarzschild black hole (continuous line) and the
luminosity flux from accretion disks inside the Bonnor solution (dashed
line) with h = −0.000214. The value of the constant h has been chosen
in order to have the mass of the Bonnor solution between r0 and infinity
(given by Eq. (12)) equal to the Schwarzschild mass M as measured
by observers at infinity. For this particular plot the chosen value of r0
was r0 = 0, 003M. Bottom panel Comparison of the luminosity flux
in the Bonnor space-time (dashed line) with a Kerr black hole with
a = 0.9 (continuous line). The value of the ISCO for such a black hole
is r 8.7M. Note that the luminosity depends on the location of the
ISCO and therefore on the value of a; however, it is generally lower
than the Bonnor case. For h, M and r0 the values chosen are the same
as the ones in the top panel
emits like a black body locally and a multi-color black body
when integrated radially. The corona is a hot (∼100 keV),
usually optically thin, electron cloud enshrouding the disk.
Due to inverse Compton scattering of some thermal
photons of the disk off the hot electrons in the corona, the latter
becomes an X-ray source with a power-law spectrum. A
fraction of X-ray photons can illuminate back the accretion disk,
producing a reflection component with some fluorescence
emission lines. The most prominent feature in the reflection
spectrum is the iron Kα line. In the case of neutral iron, this
line is at 6.4 keV, but it can shift up to 6.97 keV in the case of
H-like iron ions (which can be the case in the accretion disk
of a stellar-mass black hole). For a review, see e.g. Ref. [
33
].
The study of the shape of the iron line can be a powerful
tool to probe the space-time metric around black hole
candidates and test the nature of these compact objects [
9–12
].
The exact shape of the iron line is determined by the
metric of the space-time, the geometry of the emitting region in
the accretion disk, the intensity profile, and the inclination
angle of the disk with respect to the line of sight of the distant
observer. The choice of the intensity profile is crucial in the
final measurement, but current studies usually assume a
simple power law, namely the local intensity is Ie ∝ 1/r q where
the emissivity index q is a free parameter to be determined by
the fit. A slightly more sophisticated model is a broken power
law, where Ie ∝ 1/r q1 for r < rb, Ie ∝ 1/r q2 for r > rb, and
there are three free parameters, the two emissivity indices q1
and q2 and the breaking radius rb.
The calculations of the shape of the iron line in a generic
stationary, axisymmetric, and asymptotically flat space-time
have been already extensively discussed in the literature. In
our case, we use the code described in [
34, 35
]. The accretion
disk is described by the Novikov–Thorne model, where the
disk is in the equatorial plane perpendicular to the object’s
spin and the particles of the gas follow nearly geodesic
circular orbit in the equatorial plane. The inner edge of the disk is
at the ISCO radius. In the Bonnor space-time there is no ISCO
and therefore we set the inner edge at some arbitrary radius
rin. Some examples of iron Kα line in the reflection
spectrum of an accretion disk in Bonnor space-times are shown in
Fig. 3, where rin = 1. An accretion disk only exists for h < 0
Fig. 4 Top panel simulated
data and best fit of the observed
spectrum assuming that the
source is a bright AGN. Bottom
panel ratio between the
simulated data and the best fit.
The spectrum of the source is
computed assuming the Bonnor
metric with h = −0.1. The
viewing angle is i = 45◦ and the
exposure time is τ = 1 Ms. The
minimum of the reduced χ2 is
about 1.07. See the text for more
details
Energy (keV)
5
and the iron line becomes broader as h decreases. In both
panels in Fig. 3, we assume the intensity profile Ie ∝ 1/r q ,
where q = 3 in the top panel and q = 5 in the bottom panel.
As we can see, the intensity profile plays an important role
in the final shape of the line.
In order to understand whether current X-ray observations
of AGN can already rule out, or constrain, the possibility
that the metric around the supermassive compact objects in
galactic nuclei can be described by the Bonnor solution
discussed in the present paper, we follow the strategy already
employed in Refs. [
36–38
]. The reflection spectra in current
X-ray data are commonly fitted with Kerr models and there is
no tension between predictions and observational data. This
means that, even if the space-time metric around AGN were
not described by the Kerr solution, current data would not
be able to unambiguously identify deviations from the Kerr
space-time. We thus simulate observations with a current
Xray mission employing iron lines computed in the Bonnor
metric and we fit the simulations with a Kerr model. If the fit
is acceptable, we conclude that current data cannot exclude
the possibility that Bonnor metrics describe the space-time
around AGN. In the opposite case, if the fit is bad, we argue
that current data can rule out the Bonnor metric of the
simulation.
As theoretical model for the simulations, we employ a
simple power law with photon index = 2 to describe the
spectrum of the hot corona and a single iron Kα line to describe
the reflection spectrum of the disk. We assume the typical
parameters of a bright AGN with a strong iron line. The flux
in the energy range 0.7–10 keV is about 2 × 10−10 erg/s/cm2
and the iron line equivalent width is about 200 eV. We
convolve this spectrum with the response of the instrument in
order to get the simulated observation. The simulations are
done considering the detector XIS0 on board of the X-ray
mission Suzaku and assuming that the expose time is 1 Ms.
Our simulations show that we can always obtain good fits
for any value of h. The crucial point is the emissivity index
q. At the moment, we do to know the exact geometry of the
corona, and therefore we cannot predict the exact intensity
profile. For example, if we employ q = 3 in the simulations,
our iron line cannot be fitted with a Kerr model. However, if
we use q = 5, we alway find acceptable fits. An example is
shown in Fig. 4, where we see (bottom panel) that the ratio
between simulated data and the Kerr model is always close
to 1. We can thus assert that current observational facilities
are likely unable to test the Bonnor metric in AGN. The
limitation is in the current photon count. For example, if we
simulate the observation of a stellar-mass black hole, namely
we assume that the energy flux is 2 × 10−9 erg/s/cm2, we see
the difference, as shown in Fig. 5. Future X-ray missions,
with a larger effective area, can thus test the Bonnor metric
in AGN.
5 Discussion
Strong gravitational fields generated by axially symmetric
rotating matter sources can be of importance in astrophysics,
especially when it comes to understanding the supermassive
compact objects that dwell at the center of galaxies. In the
present article, we have considered a standard way to test
the assumption that such objects must be Kerr black holes
by comparing the properties of accretion disks around black
holes with those of some space-time metric that does not
describe a black hole.
Our main assumption is that the galactic dark matter halos
require a relativistic description towards the galactic center
and that the Bonnor space-time (describing rigidly rotating
dust) may provide a good first approximation for a relativistic
model of dark matter. The most commonly used models for
dark matter, like for example the Jaffe density profile [
29
] or
the NFW profile [
28
], present a divergence of the density at
the center while at the same time being purely Newtonian. It
is then natural to assume that relativistic effects will become
important in regions where the density becomes high. These
regions are also the centers of galaxies where we believe that
supermassive black holes reside. Therefore we have
considered a simple relativistic space-time describing rotating dust
and suggested the possibility that this kind of matter
distributions, together with the effects of general relativity, may be
sufficient to explain the behaviour of accretion disks near the
galactic centers, possibly without invoking the presence of
supermassive black holes. To this aim we have constructed
a simple observational test that in the near future could be
used to check the validity of such models.
We have shown that the effective potential for ordinary
massive particles moving in the accretion disk of the
Bonnor solution is considerably different from that of black hole
solutions. As a consequence, the luminosity spectrum of the
accretion disk in such a space-time will also differ
substantially from the spectrum of accretion disks around black
holes. The shape of the iron Kα line in the reflection spectrum
is also substantially different and it can be used to test this
scenario with future X-ray missions. Future observations of
the shadow of the supermassive compact object at the
galactic center will soon allow us to test the assumption that such
object must be a black hole.
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1. J. Kormendy , D. Richstone, Ann. Rev. Astron. Astrophys. 33 , 581 ( 1995 )
2. C. Bambi , Rev. Mod. Phys . 89 ( 2 ), 025001 ( 2017 )
3. A.N. Chowdhury , M. Patil , D. Malafarina , P.S. Joshi , Phys. Rev. D 85 , 104031 ( 2012 )
4. D. Pugliese , H. Quevedo , R. Ruffini , Phys. Rev. D 84 , 044030 ( 2011 )
5. Z. Kovacs , T. Harko, Phys. Rev. D 82 , 124047 ( 2010 )
6. P.S. Joshi , D. Malafarina , R. Narayan , Class. Quantum Gravity 28 , 235018 ( 2011 )
7. P.S. Joshi , D. Malafarina , R. Narayan , Class. Quantum Gravity 31 , 015002 ( 2014 )
8. K.S. Virbhadra , G.F.R. Ellis , Phys. Rev. D 65 , 103004 ( 2002 )
9. J. Jiang , C. Bambi , J.F. Steiner , JCAP 1505 , 025 ( 2015 )
10. J. Jiang , C. Bambi , J.F. Steiner , Astrophys. J. 811 , 130 ( 2015 )
11. Y. Ni , J. Jiang , C. Bambi , JCAP 1609 , 014 ( 2016 )
12. C. Bambi , A. Cardenas-Avendano , T. Dauser , J.A. Garcia , S. Nampalliwar , Astrophys. J. 842 , 76 ( 2017 )
13. C. Bambi , D. Malafarina , Phys. Rev. D 88 , 064022 ( 2013 )
14. W.B. Bonnor , J. Phys . A Math. Gen . 10 , 1673 ( 1977 )
15. R. Opher , N.O. Santos , A. Wang , J. Math. Phys 37 , 1982 ( 1996 )
16. W.B. Bonnor Gen . Relativ. Gravit . 24 , 551 ( 1992 )
17. R.O. Hansen , J. Winicour , J. Math. Phys. 16 , 804 ( 1975 )
18. J. Winicour , J. Math. Phys. 16 , 1806 ( 1975 )
19. K. Lanczos , Gen. Relativ. Gravit. 29 , 363 ( 1997 ) (this is a reprint and translation of the original article which appeared in Zeitschrift fur Physik 21 , 73 ( 1927 ))
20. W.J. Van Stockum , Proc. R. Soc. Edinb. A 57 , 135 ( 1937 )
21. J.N. Islam , Phys. Lett. A 120 , 119 ( 1987 )
22. D. Malafarina , Universe 3 , 48 ( 2017 )
23. N. Itoh , Prog. Theor. Phys . 44 , 291 ( 1970 )
24. F.E. Schunck , E.W. Mielke , Class. Quantum Gravity 20 , R301 ( 2003 )
25. P.O. Mazur , E. Mottola, arXiv:gr-qc/ 0109035
26. C. Rovelli , F. Vidotto , Int. J. Mod. Phys. D 23 , 1442026 ( 2014 )
27. G. Neugebauer, R. Meinel , Astrophys. J. 414 , L97 ( 1993 )
28. J.F. Navarro , C.S. Frenk , S.D.M. White , Astrophys. J. 462 , 563 ( 1996 )
29. W. Jaffe, Mon. Not. R. Astron . Soc. 202 , 995 ( 1983 )
30. J. Gunn , J.R. Gott , Astrophys. J. 176 , 1 ( 1972 )
31. G. Castignani, N. Frusciante , D. Vernieri , P. Salucci , Nat. Sci. 4 , 265 ( 2012 )
32. J.C.N. de Araujo , A. Wang , Gen. Relativ. Gravit. 32 , 1971 ( 2000 )
33. A.C. Fabian , K. Iwasawa , C.S. Reynolds , A.J. Young , Publ. Astron. Soc. Pac . 112 , 1145 ( 2000 )
34. C. Bambi , Astrophys. J. 761 , 174 ( 2012 )
35. C. Bambi, Phys. Rev. D 87 , 023007 ( 2013 )
36. M. Zhou , A. Cardenas-Avendano , C. Bambi , B. Kleihaus , J. Kunz , Phys. Rev. D 94 , 024036 ( 2016 )
37. Y. Ni , M. Zhou , A. Cardenas-Avendano , C. Bambi , C.A.R. Herdeiro , E. Radu, JCAP 1607 , 049 ( 2016 )
38. Z. Cao , A. Cardenas-Avendano , M. Zhou , C. Bambi , C.A.R. Herdeiro , E. Radu, JCAP 1610 , 003 ( 2016 )